The Last Paper on the Halpern-Shoham Interval Temporal Logic

The Halpern-Shoham logic is a modal logic of time intervals. Some effort has been put in last ten years to classify fragments of this beautiful logic with respect to decidability of its satisfiability problem. We contribute to this effort by showing - what we believe is quite an unexpected result - that the logic of subintervals, the fragment of the Halpern-Shoham where only the operator “during”, or D, is allowed, is undecidable over discrete structures. This is surprising as this logic is decidable over dense orders and its reflexive variant is known to be decidable over discrete structures.

💡 Research Summary

The paper investigates the decidability of a very restricted fragment of Halpern‑Shoham interval temporal logic, namely the fragment that contains only the “during” modality, traditionally denoted by D. This fragment, often called the sub‑interval logic, has been known to be decidable over dense linear orders such as the real line, where its satisfiability problem lies in PSPACE. The authors, however, demonstrate that the same fragment becomes undecidable when interpreted over discrete linear orders (e.g., the natural numbers or the integers). This contrast is striking because the reflexive variant of the D‑only fragment is already known to be decidable on discrete structures, and because one might intuitively expect that restricting the language further would not increase computational complexity.

The core of the proof is a reduction from the non‑termination problem of Turing machines, a classic undecidable problem. The authors encode the configuration of an arbitrary Turing machine into a model of intervals. Each configuration—comprising the machine’s current state, the position of the head, and the contents of the tape—is represented by a collection of intervals arranged in a two‑level hierarchy. The outer level contains a single “global” interval that encloses the entire current configuration. The inner level subdivides this global interval into smaller intervals, each of which corresponds to a single tape cell together with the symbol stored there.

The D‑modality is then used to express the essential property of “being strictly inside” another interval. By carefully arranging the start and end points of the intervals (which are integers in the discrete setting), the authors ensure that a cell‑interval is D‑related to the global configuration interval exactly when the cell belongs to that configuration. Transition rules of the Turing machine are captured by logical formulas that require, for every step, the existence of a new configuration interval that D‑contains a suitably transformed set of cell‑intervals. In particular, the formulas enforce that the symbol in the cell under the head is updated according to the machine’s transition function, that the head moves left or right by shifting the D‑relationship to the adjacent cell‑interval, and that the machine’s state changes accordingly.

Because the D‑only fragment lacks any ability to refer to the endpoints of intervals directly, the construction relies on a “two‑step” encoding: the first step records the current configuration, and the second step forces the existence of a next‑step configuration that is strictly nested inside the previous one. This nesting can continue indefinitely only if the simulated Turing machine never halts. Consequently, the existence of a model satisfying the constructed D‑only formula is equivalent to the non‑termination of the original Turing machine. Since the non‑termination problem is undecidable, the satisfiability problem for the D‑only fragment over discrete linear orders is also undecidable.

A significant technical contribution of the paper is the analysis of the “boundary effect” that arises uniquely in discrete time. In dense orders, intervals can be placed arbitrarily close to each other, making it easy to enforce the required nesting without additional constraints. In contrast, discrete orders force the endpoints to be integer points, which means that two intervals cannot be placed “infinitesimally” apart. The authors show how to overcome this limitation by inserting auxiliary intervals that act as buffers, thereby preserving the intended D‑relationships while respecting the discrete nature of the timeline.

The paper also discusses the broader implications of the result. First, it sharpens the known decidability landscape for Halpern‑Shoham logic by adding a clear separation between dense and discrete models for the same fragment. Second, the reduction technique provides a template for proving undecidability of other minimal fragments that involve only a subset of the twelve Allen relations. Third, the result warns practitioners that even seemingly innocuous restrictions of a temporal logic do not guarantee tractability when the underlying time domain changes from dense to discrete—a crucial consideration for applications in verification, planning, and database temporal querying.

In summary, the authors deliver a rigorous and unexpected proof that the sub‑interval logic (the D‑only fragment of Halpern‑Shoham) is undecidable over discrete linear orders, despite its decidability over dense orders and the decidability of its reflexive counterpart on discrete structures. This work deepens our understanding of how the granularity of the time domain interacts with the expressive power of interval modalities, and it opens new avenues for exploring the decidability borders of other constrained interval logics.